A solution to Pfeffer's problem (Q1852436)

On the square \(S=[0,1]\times [0,1]\) a function \(f\) is given for which \[ \int_a^b \left(\int_c^d f(x,y) dy\right) dx = \int_c^d\left(\int_a^b f(x,y) dx\right)dy \] for each subinterval \([a,b]\times [c,d] \subset S\) while the integral \(\int_S f\) does not exist. The above equality can be taken assuming that the integrals are Lebesgue integrals. The function \(f\) constructed in the paper is even not Kurzweil integrable on the square \(S\).